To solve this question, we need to understand the characteristics of a binomial distribution given specific parameters. The parameters of the binomial distribution provided are [tex]\( n = 6 \)[/tex] (number of trials) and [tex]\( p = 0.9 \)[/tex] (probability of success on each trial).
### Step-by-Step Solution:
1. Determine the Center of the Distribution:
The center of a binomial distribution, also known as the mean ([tex]\(\mu\)[/tex]), is calculated using the formula:
[tex]\[
\mu = n \times p
\][/tex]
Here, [tex]\( n = 6 \)[/tex] and [tex]\( p = 0.9 \)[/tex].
Plugging in the values:
[tex]\[
\mu = 6 \times 0.9 = 5.4
\][/tex]
Therefore, the center of the distribution is [tex]\( 5.4 \)[/tex].
2. Determine the Shape of the Distribution:
The shape of a binomial distribution depends on the values of [tex]\( p \)[/tex] and [tex]\( (1 - p) \)[/tex]:
- When [tex]\( p = 0.5 \)[/tex], the distribution is approximately symmetric.
- When [tex]\( p \)[/tex] is significantly greater than [tex]\( 0.5 \)[/tex], the distribution tends to be skewed left (more to the right).
- When [tex]\( p \)[/tex] is significantly less than [tex]\( 0.5 \)[/tex], the distribution tends to be skewed right (more to the left).
In this case, [tex]\( p = 0.9 \)[/tex] means the probability of success is very high, and the probability of failure [tex]\( (1 - p) = 0.1 \)[/tex] is very low.
This leads to a situation where most of the outcomes are clustered towards the higher number of successes, resulting in a distribution that is skewed left.
Combining these observations:
- The center (mean) of the distribution is [tex]\( 5.4 \)[/tex].
- The shape of the distribution is skewed left.
Thus, the correct answer is:
```
Center: 5.4
Shape: skewed left
```
